Multimodal Dependent Type Theory

TL;DR

This paper introduces MTT, a flexible dependent type theory supporting multiple modalities, capable of modeling various modal systems and unifying prior theories while also discovering new ones, with proven consistency and canonicity.

Contribution

MTT provides a unified, parametrized framework for multiple modal dependent type theories, simplifying their syntax and enabling new instantiations that improve existing systems.

Findings

Reproduces prior modal type theories such as guarded recursion and cohesion.

Uncovers previously unknown type theories through specific mode theory instantiations.

Proves the consistency and canonicity of MTT across all mode theories.

Abstract

We introduce MTT, a dependent type theory which supports multiple modalities. MTT is parametrized by a mode theory which specifies a collection of modes, modalities, and transformations between them. We show that different choices of mode theory allow us to use the same type theory to compute and reason in many modal situations, including guarded recursion, axiomatic cohesion, and parametric quantification. We reproduce examples from prior work in guarded recursion and axiomatic cohesion, thereby demonstrating that MTT constitutes a simple and usable syntax whose instantiations intuitively correspond to previous handcrafted modal type theories. In some cases, instantiating MTT to a particular situation unearths a previously unknown type theory that improves upon prior systems. Finally, we investigate the metatheory of MTT. We prove the consistency of MTT and establish canonicity through an extension of recent type-theoretic gluing techniques. These results hold irrespective of the choice of mode theory, and thus apply to a wide variety of modal situations.